## Mission

Despite the development of increasingly efficient instruments, some physical constraints limit the achievable performance. For example, at a given frequency, the angular resolution of an antenna is limited by its size.

A way to overcome the various constraints inherent in the construction of large instruments is to cut a large instrument into smaller instruments. A famous example is the Event Horizon Telescope, which is made up of several radio telescopes spread all over the Earth.

In the space field, the constraints induced by the production of large instruments are strong and the largest instruments generally do not exceed a few tens of meters. To make larger instruments a solution is to distribute the instrument (typically an antenna) over a set of remote satellites, each performing part of the measurement. The reconstruction of the final measurement involves a step of combining individual measurements which requires detailed knowledge of the position of the various measurement points and their dating.

For interferometry applications or in general, when it comes to reconstitute an antenna from unit elements, the frequency band observed imposes the constraints of location and synchronization of the satellites: the higher this frequency, the more precise it is necessary to know the position / synchronization of the different satellites. The positioning constraints can be expressed in the first order as a fraction of the wavelength, which means that an accuracy better than the centimeter is needed as soon as the frequency goes beyond 300 MHz.

This kind of precision requires techniques based on carrier phase measurement. These methods are already used in the terrestrial environment on GNSS signals. But in the case of GNSS satellites, atomic clocks on board the satellites are used to generate signals of excellent quality and advanced techniques of orbit restitution and correction of various biases are implemented in terrestrial calculation centers. When considering a swarm of satellites, one generally considers satellites of small size and therefore unable to carry atomic clocks as efficient as those used in GNSS satellites. A promising research track to improve the estimation of the absolute position of all satellites is to resort to techniques exploiting the associated Euclidean Distance Matrix (EDM) [1], that is inter-satellites distances. To be effective, these techniques must exploit distance measurement as precise as possible and measurement of the absolute positions of a subset of satellites of the swarm.

Thus, achieving precise distance measurements based on carrier phase measurements definitely represent a critical prerequisite to resort to EDM based algorithms in the case of a swarm of satellites.

The main objective of this thesis would therefore be to study the feasibility of a system for measuring the distance between satellites of a swarm with centimeter precision, on the basis of the measurement of the phase of a carrier signal. The lack of a mean of precise absolute positioning would be one of the main constraints.

The use of a carrier signal dedicated to this distance measurement or the use of an opportunity signal such as that implemented by a data link between satellites will be considered. In both cases, the thesis will determine the minimum signal specifications for the positioning need.

Phase measurement inherently provides a measure of the distance modulo the wavelength of the carrier signal. This ambiguity must be removed. The thesis will assess whether the processes conventionally used for this task, based on the integer least squares method or the Kalman filter, can be applied in the particular study case of the thesis.

The continuity and availability of the distance measurement will also be part of the thesis. Indeed, it must be considered that the satellites of the swarm are not controlled in attitude and their free rotation could cause periodic discontinuities in the reception of the carrier signal. Consequently, the phase measurement would not be continuous either and the delays for resolving ambiguities after the reacquisition of the signal would also have an effect on the availability of the measurement. This thesis will characterize these limits of the system.

If time permits, the thesis will study to what extent it would be possible, by coupling with an attitude model of the satellite, to propagate the phase during the carrier signal cut-off periods to provide a continuous measurement. One of the answers expected from this part would be to know if a minimum attitude control system would be necessary, to keep the speed of rotation of the satellite within certain limits, for example.

[1] I. Dokmanic, R. Parhizkar, J. Ranieri, M. Vetterli, Euclidean Distance Matrices. Essential theory, algorithms, and applications, IEEE Signal Processing Magazine, 32(6), 2015

For more Information, contact Directeur de thèse : **Eric.CHAUMETTE@isae-supaero.fr **

**about the topics and the co-financial partner (found by the lab !). Then, prepare a resumé, a recent transcript and a reference letter from your M2 supervisor/ engineering school director and you will be ready to apply online !**

## Profil

# Message from PhD Team

CNES will inform about the status of your application in mid-June. More details on CNES website : https://cnes.fr/en/web/CNES-en/10685-st-doctoral-grants.php